CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.3
Traction vector t i (n)ˆ acting at point P of plane element ∆S i, whose normal is n i.
and
lim *
∆S
→0
∆Mi
= 0 *
∆S
(3.2-2)
The vector dfi/dS* = t
( ˆn ) is called the stress vector, or sometimes the traction
i
vector. In Eq 3.2-2 we have made the assumption that in the limit at P the
moment vector vanishes, and there is no remaining concentrated moment,
or couple stress as it is called.
The appearance of ( ˆn ) in the symbol t
( ˆn ) for the stress vector serves to remind
i
us that this is a special vector in that it is meaningful only in conjunction with
its associated normal vector ˆn at P. Thus, for the infinity of cutting planes
imaginable through point P, each identified by a specific ˆn , there is also an
infinity of associated stress vectors for a given loading of the body. The
totality of pairs of the companion vectors and at P, as illustrated by a
typical pair in Figure 3.3, defines the state of stress at that point.
By applying Newton’s third law of action and reaction across the cutting
plane, we observe that the force exerted by Portion I upon Portion II is equal
and opposite to the force of Portion II upon Portion I. Additionally, from the
principle of linear momentum (Newton’s second law) we know that the time
rate of change of the linear momentum of any portion of a continuum body
is equal to the resultant force acting upon that portion. For Portions I and
II, this principle may be expressed in integral form by the respective equations
(these equations are derived in Section 5.4 from the principle of linear
momentum),
t ( ˆn )
i
t
( ˆn ) ˆn
i
t dS+ bdV = d
( ˆn )
i ρ i ρvdV
∫ ∫
i
V dt ∫V
SI I
I
t dS+ bdV = d
( ˆn )
i ρ i ρvdV
∫ ∫
i
V dt ∫V
SII II
II
(3.2-3a)
(3.2-3b)